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Decidable theories of pseudo-$$p$$-adic closed fields. (English. Russian original) Zbl 0717.12005
Algebra Logic 28, No. 6, 421-438 (1989); translation from Algebra Logika 28, No. 6, 643-669 (1989).
Let $$K$$ a field and $$\Gamma$$ an ordered Abelian group. A valuation $$\phi: K\to \Gamma \cup \{\infty \}$$ is called $$p$$-valuation if $$\text{char}(K)=0$$, $$\phi(p)$$ is the least element of $$\Gamma$$ greater than zero, and the residue class field is the $$p$$-element field. A field $$K$$ is called P$$p$$C-field if $$\text{char}(K)=0$$ and $$K$$ is existentially closed in every regular totally $$p$$-adic extension of $$K$$. We recall that the theory of the class of P$$p$$C-fields is undecidable. The author proves that the theory of maximal P$$p$$C-fields and the theories of the classes of P$$p$$C-fields with finitely generated absolute Galois groups are decidable.

##### MSC:
 12L05 Decidability and field theory 03B25 Decidability of theories and sets of sentences 12J12 Formally $$p$$-adic fields
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##### References:
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